Definite Integration
GIF Integral + Trig Integral via King's Rule
nta_pyq_2026_jan
Grade 12
Question:
Let $[\cdot]$ be the greatest integer function. If $\alpha=\displaystyle\int_0^{64}\left(x^{1/3}-\left[x^{1/3}\right]\right)dx$, then $\dfrac{1}{\pi}\displaystyle\int_0^{\alpha\pi}\left(\frac{\sin^2\theta}{\sin^6\theta+\cos^6\theta}\right)d\theta$ is equal to _____
Step-by-Step Solution
Key Concept: $\int_0^{64}x^{1/3}dx=\frac{3}{4}[x^{4/3}]_0^{64}=192$. $\int_0^{64}[x^{1/3}]dx=\sum_{k=0}^{3}\int_{k^3}^{(k+1)^3}k\,dx=0+1(8-1)+\ldots=156$. So $\alpha=36$.
$\alpha=36$. $E=36$.
Correct Answer: 36